Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2011, Article ID 814132, 8 pages
doi:10.1155/2011/814132
Research Article
New Method for Solving Linear Fractional
Differential Equations
S. Z. Rida and A. A. M. Arafa
Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt
Correspondence should be addressed to A. A. M. Arafa, anaszi2@yahoo.com
Received 4 May 2011; Revised 21 July 2011; Accepted 25 July 2011
Academic Editor: Shaher M. Momani
Copyright q 2011 S. Z. Rida and A. A. M. Arafa. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We develop a new application of the Mittag-Leffler Function method that will extend the
application of the method to linear differential equations with fractional order. A new solution
is constructed in power series. The fractional derivatives are described in the Caputo sense. To
illustrate the reliability of the method, some examples are provided. The results reveal that the
technique introduced here is very effective and convenient for solving linear differential equations
of fractional order.
1. Introduction
Fractional differential equations have excited, in recent years, a considerable interest both
in mathematics and in applications. They were used in modeling of many physical and
chemical processes and engineering see, e.g., 1–6. In its turn, mathematical aspects of
fractional differential equations and methods of their solutions were discussed by many
authors: the iteration method in 7, the series method in 8, the Fourier transform technique
in 9, 10, special methods for fractional differential equations of rational order or for
equations of special type in 11–16, the Laplace transform technique in 3–6, 16, 17,
and the operational calculus method in 18–23. Recently, several mathematical methods
including the Adomian decomposition method 18–25, variational iteration method 23–26
and homotopy perturbation method 27, 28 have been developed to obtain the exact and
approximate analytic solutions. Some of these methods use transformation in order to reduce
equations into simpler equations or systems of equations, and some other methods give the
solution in a series form which converges to the exact solution.
The reason of using fractional order differential FOD equations is that FOD equations
are naturally related to systems with memory which exists in most biological systems. Also
they are closely related to fractals which are abundant in biological systems. The results
2
International Journal of Differential Equations
derived from the fractional system are of a more general nature. Respectively, solutions to the
fractional diffusion equation spread at a faster rate than the classical diffusion equation and
may exhibit asymmetry. However, the fundamental solutions of these equations still exhibit
useful scaling properties that make them attractive for applications.
The concept of fractional or noninteger order derivation and integration can be traced
back to the genesis of integer order calculus itself 29. Almost all of the mathematical theory
applicable to the study of noninteger order calculus was developed through the end of the
19th century. However, it is in the past hundred years that the most intriguing leaps in
engineering and scientific application have been found. The calculation techniques in some
cases meet the requirement of physical reality. The use of fractional differentiation for the
mathematical modeling of real-world physical problems has been widespread in recent years,
for example, the modeling of earthquake, the fluid dynamic traffic model with fractional
derivatives, and measurement of viscoelastic material properties. Applications of fractional
derivatives in other fields and related mathematical tools and techniques could be found in
30–41. In fact, real-world processes generally or most likely are fractional order systems.
The derivatives are understood in the Caputo sense. The general response expression
contains a parameter describing the order of the fractional derivative that can be varied to
obtain various responses.
2. Fractional Calculus
There are several approaches to the generalization of the notion of differentiation to fractional
orders, for example, the Riemann-Liouville, Grünwald-Letnikov, Caputo, and generalized
functions approach 42. The Riemann-Liouville fractional derivative is mostly used by
mathematicians but this approach is not suitable for real-world physical problems since
it requires the definition of fractional order initial conditions, which have no physically
meaningful explanation yet. Caputo introduced an alternative definition, which has the
advantage of defining integer order initial conditions for fractional order differential
equations 42. Unlike the Riemann-Liouville approach, which derives its definition from
repeated integration, the Grünwald-Letnikov formulation approaches the problem from the
derivative side. This approach is mostly used in numerical algorithms.
Here, we mention the basic definitions of the Caputo fractional-order integration and
differentiation, which are used in the upcoming paper and play the most important role in
the theory of differential and integral equation of fractional order.
The main advantages of Caputo approach are the initial conditions for fractional
differential equations with the Caputo derivatives taking on the same form as for integer
order differential equations.
Definition 2.1. The fractional derivative of fx in the Caputo sense is defined as
Dα fx I m−α Dm fx
x
1
x − tm−α1 f m tdt
Γm − α 0
for m − 1 < α ≤ m, m ∈ N, x > 0.
2.1
International Journal of Differential Equations
3
For the Caputo derivative we have Dα C 0, C is constant,
D α tn ⎧
⎪
0,
⎪
⎨
⎫
⎪
n ≤ α − 1, ⎪
⎬
Γn 1 n−α
.
⎪
t , n > α − 1. ⎪
⎪
⎪
⎩ Γn − α 1
⎭
2.2
Definition 2.2. For m to be the smallest integer that exceeds α, the Caputo fractional derivative
of order α > 0 is defined as
∂α ux, t
∂tα
⎫
⎧
m
t
1
m−α1 ∂ ux, τ
⎪
⎪
⎪
⎪
dτ,
for
m
−
1
<
α
<
m
−
τ
t
⎬
⎨
Γm − α 0
∂τ m
.
m
⎪
⎪
⎪
⎪
⎭
⎩ ∂ ux, t ,
for
α
m
∈
N
∂tm
Dα ux, t 2.3
3. Analysis of the Method
The Mittag-Leffler 1902–1905 functions Eα and Eα,β 42, defined by the power series
Eα z ∞
zk
,
Γαk 1
k0
Eα,β z ∞
zk
,
k0 Γ αk β
α > 0, β > 0,
3.1
have already proved their efficiency as solutions of fractional order differential and integral
equations and thus have become important elements of the fractional calculus theory and
applications.
In this paper, we will explain how to solve some of differential equations with
fractional level through the imposition of the generalized Mittag-Leffler function Eα z. The
generalized Mittag-Leffler method suggests that the linear term yx is decomposed by an
infinite series of components:
y Eα axα ∞
n0
an
xnα
.
Γnα 1
3.2
We will use the following definitions of fractional calculus:
Dα y ∞
an
n1
D2α y ∞
an
n2
xn−1α
,
Γn − 1α 1
3.3
xn−2α
.
Γn − 2α 1
3.4
This is based on the Caputo fractional is derivatives. The convergence of the Mittag Leffler
function discussed in 42.
4
International Journal of Differential Equations
4. Applications and Results
In this section, we consider a few examples that demonstrate the performance and efficiency
of the generalized Mittag-Leffler function method for solving linear differential equations
with fractional derivatives.
Example 4.1. Consider the following fractional differential equation 43:
dα y
Ay.
dxα
4.1
By using 3.3 into 4.1 we find
∞
an
n1
∞
xn−1α
xnα
− A an
0.
Γn − 1α 1
Γnα 1
n0
4.2
Combining the alike terms and replacing n by n 1 in the first sum, we assume the form
∞
an1
n0
∞
xnα
xnα
− A an
0,
Γnα 1
Γnα 1
n0
∞ n1
a
− Aa
n
n0
xnα
0.
Γnα 1
4.3
With the coefficient of xnα equal to zero and identifying the coefficients, we obtain recursive
an1 − Aan 0 ⇒ an1 Aan ,
at n 0,
a1 Aa0 A,
at n 1,
a2 Aa1 ⇒ a2 A2 ,
at n 2,
a3 Aa2 ⇒ a3 A3 .
4.4
Substituting into 3.2
yx a0 a1
xα
x2α
x3α
a2
a3
··· ,
Γα 1
Γ2α 1
Γ3α 1
xα
x2α
x3α
yx 1 A
A2
A3
··· .
Γα 1
Γ2α 1
Γ3α 1
4.5
The general solution is
yx ∞
An xnα
.
Γnα 1
n0
4.6
International Journal of Differential Equations
5
We can write the general solution in the Mittag-Leffler function form as
yx Eα An xα .
4.7
As α 1, we have the exact solution:
yx ∞
Axn
eAx ,
Γn
1
n0
4.8
which is the exact solution of the standard form.
Example 4.2. Consider the fractional differential equation 44
d2α y
− y 0.
dx2α
4.9
By using 3.2 and 3.4 into 4.9 we find
∞
an
n2
∞
xn−2α
xnα
− an
0.
Γn − 2α 1 n0 Γnα 1
4.10
Combining the alike terms and replacing n by n 2 in the first sum, we assume the form
∞
an2
n0
∞
xnα
xnα
− an
0,
Γnα 1 n0 Γnα 1
∞ an2 − an
n0
xnα
0.
Γnα 1
4.11
With the Coefficient of xnα equal to zero and identifying the coefficients, we obtain recursive
an2 an .
4.12
Substituting into 3.2, we find that:
yx 1 a
x2α
x3α
xα
a2
··· .
Γα 1 Γ2α 1
Γ3α 1
4.13
If a 1, we can write the general solution in the Mittag-Leffler function form as
yx ∞
xα
Eα xα Γnα 1
n0
which is the exact solution of the linear fractional differential equation 4.9.
4.14
6
International Journal of Differential Equations
Example 4.3. Consider the fractional differential equation 43
d2α y dα y
− 2y 0.
dx2α dxα
4.15
By using 3.2 and 3.4 into 4.15 we find
∞
n2
an
∞
∞
xn−2α
xn−1α
xnα
an
− 2 an
0.
Γn − 2α 1 n1 Γn − 1α 1
Γnα 1
n0
4.16
Combining the alike terms and replacing n by n 2 in the first sum, we assume the form
∞
n0
an2
∞
∞
xnα
xnα
xnα
an1
− 2 an
0,
Γnα 1 n0
Γnα 1
Γnα 1
n0
∞ n2
a
a
n1
n0
− 2a
n
xnα
0
Γnα 1
4.17
With the coefficient of xnα equal to zero and identifying the coefficients, we obtain recursive
an2 2an − an1 .
4.18
Substituting into 3.2, we find that:
yx 1 a
x2α
x3α
xα
2 − a
a − 2
··· .
Γα 1
Γ2α 1
Γ3α 1
4.19
If a 1, we can write the general solution in the Mittag-Leffler function form as
yx ∞
xα
Eα xα Γnα 1
n0
4.20
which is the solution of the linear fractional differential equation 4.15.
5. Conclusions
A new generalization of the Mittag-Leffler function method has been developed for linear
differential equations with fractional derivatives. The new generalization is based on the
Caputo fractional derivative. It may be concluded that this technique is very powerful and
efficient in finding the analytical solutions for a large class of linear differential equations of
fractional order.
International Journal of Differential Equations
7
References
1 Y. I. Babenko, Heat and Mass Transfer, Chemia, Leningrad, Germany, 1986.
2 M. Caputo and F. Mainardi, “Linear models of dissipation in anelastic solids,” La Rivista del Nuovo
Cimento, vol. 1, no. 2, pp. 161–198, 1971.
3 R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional
order,” in Fractals Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainar, Eds., pp.
223–276, Springer, New York, NY, USA, 1997.
4 R. Gorenflo and R. Rutman, “On ultraslow and intermediate processes,” in Transform Methods and
Special Functions, P. Rusev, I. Dimovski, and V. Kiryakova, Eds., pp. 61–81, Science Culture Technology
Publishing, Singapore, 1995.
5 F. Mainardi, “Fractional relaxation and fractional diffusion equations, mathematical aspects,” in
Proceedings of the 12th IMACS World Congress, W. F. Ames, Ed., vol. 1, pp. 329–332, Georgia Tech
Atlanta, 1994.
6 F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in
Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., pp. 291–
348, Springer, New York, NY, USA, 1997.
7 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and
Applications, Gordon and Breach Science Publishers, New York, NY, USA, 1993.
8 H. Beyer and S. Kempfle, “Definition of physically consistent damping laws with fractional
derivatives,” Journal of Applied Mathematics and Mechanics, vol. 75, pp. 623–635, 1995.
9 S. Kempfle and H. Beyer, “Global and causal solutions of fractional differential equations,” in
Proceedings of the 2nd International Workshop on Transform Methods and Special Functions, pp. 210–216,
Science Culture Technology Publishing, Varna, Bulgaria, 1996.
10 R. L. Bagley, “On the fractional order initial value problem and its engineering applications,” in
Fractional Calculus and Its Applications, K. Nishimoto, Ed., pp. 12–20, College of Engineering, Nihon
University, Tokyo, Japan, 1990.
11 A. A. Kilbas and M. Saigo, “On mittag-leffler type function, fractional calculus operators and
solutions of integral equations,” Integral Transforms and Special Functions, vol. 4, no. 4, pp. 355–370,
1996.
12 Y. F. Luchko and H. M. Srivastava, “The exact solution of certain differential equations of fractional
order by using operational calculus,” Computers and Mathematics with Applications, vol. 29, no. 8, pp.
73–85, 1995.
13 M. W. Michalski, “On a certain differential equation of non-integer order,” Zeitschrift fur Analysis und
ihre Anwendungen, vol. 10, pp. 205–210, 1991.
14 M. W. Michalski, Derivatives of Non-integer Order and Their Applications, vol. 328 of Dissertationes
Mathematicae, Polska Akademia Nauk, Institut Matematyczny, Warszawa, Poland, 1993.
15 K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations,
John Wiley & Sons, New York, NY, USA, 1993.
16 I. Podlubny, “Solution of linear fractional differential equations with constant coefficients,” in
Transform Methods and Special Functions, P. Rusev, I. Dimovski, and V. Kiryakova, Eds., pp. 227–237,
Science Culture Technology Publishing, Singapore, 1995.
17 S. B. Hadid and Y. F. Luchko, “An operational method for solving fractional differential equations of
an arbitrary real order,” Pan-American Mathematical Journal, vol. 6, pp. 57–73, 1996.
18 J. Padovan, “Computational algorithms for FE formulations involving fractional operators,”
Computational Mechanics, vol. 2, no. 4, pp. 271–287, 1987.
19 S. Momani, “Non-perturbative analytical solutions of the space- and time-fractional Burgers
equations,” Chaos, Solitons and Fractals, vol. 28, no. 4, pp. 930–937, 2006.
20 S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation by
Adomian decomposition method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 488–494,
2006.
21 Z. M. Odibat and S. Momani, “Approximate solutions for boundary value problems of time-fractional
wave equation,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 767–774, 2006.
22 S. Momani, “An explicit and numerical solutions of the fractional KdV equation,” Mathematics and
Computers in Simulation, vol. 70, no. 2, pp. 110–118, 2005.
8
International Journal of Differential Equations
23 S. Momani and Z. Odibat, “Analytical approach to linear fractional partial differential equations
arising in fluid mechanics,” Physics Letters, Section A, vol. 355, no. 4-5, pp. 271–279, 2006.
24 S. Momani and Z. Odibat, “Numerical comparison of methods for solving linear differential equations
of fractional order,” Chaos, Solitons and Fractals, vol. 31, no. 5, pp. 1248–1255, 2007.
25 J. H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous
media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998.
26 Z. M. Odibat and S. Momani, “Application of variational iteration method to nonlinear differential
equations of fractional order,” International Journal of Nonlinear Sciences and Numerical Simulation, vol.
7, no. 1, pp. 15–27, 2006.
27 Z. Odibat and S. Momani, “Modified homotopy perturbation method: application to quadratic Riccati
differential equation of fractional order,” Chaos, Solitons and Fractals, vol. 36, no. 1, pp. 167–174, 2008.
28 S. Momani and Z. Odibat, “Comparison between the homotopy perturbation method and the
variational iteration method for linear fractional partial differential equations,” Computers and
Mathematics with Applications, vol. 54, no. 7-8, pp. 910–919, 2007.
29 J. D. Munkhammar, “Fractional calculus and the Taylor–Riemann series,” Undergraduate Mathematics
Journal, vol. 6, 2005.
30 R. L. Magin, “Fractional calculus in bioengineering,” Critical Reviews in Biomedical Engineering, vol. 32,
no. 1, pp. 1–104, 2004.
31 R. L. Magin, “Fractional calculus in bioengineering, part 2,” Critical Reviews in Biomedical Engineering,
vol. 32, no. 2, pp. 105–193, 2004.
32 R. L. Magin, “Fractional calculus in bioengineering, part 3,” Critical Reviews in Biomedical Engineering,
vol. 32, no. 3-4, pp. 195–377, 2004.
33 K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
34 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives-Theory and
Applications, Gordon and Breach Science Publishers, Longhorne, Pa, USA, 1993.
35 S. Z. Rida, A. M. A. El-Sayed, and A. A. M. Arafa, “Effect of bacterial memory dependent growth by
using fractional derivatives reaction-diffusion chemotactic model,” Journal of Statistical Physics, vol.
140, no. 4, pp. 797–811, 2010.
36 A. M. A. El-Sayed, S. Z. Rida, and A. A. M. Arafa, “On the Solutions of the generalized reactiondiffusion model for bacteria growth,” Acta Applicandae Mathematicae, vol. 110, pp. 1501–1511, 2010.
37 S. Z. Rida, A. M. A. El-Sayed, and A. A. M. Arafa, “On the solutions of time-fractional reactiondiffusion equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp.
3847–3854, 2010.
38 S. Z. Rida, A. M. A. El-Sayed, and A. A. M. Arafa, “A Fractional Model for Bacterial Chemoattractant
in a Liquid Medium,” Nonlinear Science Letters A, vol. 1, no. 4, pp. 415–420, 2010.
39 A. M. A. El-Sayed, S. Z. Rida, and A. A. M. Arafa, “Exact solutions of fractional-order biological
population model,” Communications in Theoretical Physics, vol. 52, no. 6, pp. 992–996, 2009.
40 A. M. A. El-Sayed, S. Z. Rida, and A. A. M. Arafa, “On the solutions of time-fractional bacterial
chemotaxis in a diffusion gradient chamber,” International Journal of Nonlinear Sciences, vol. 7, no. 4, p.
485, 2009.
41 S. Z. Rida, H. M. El-Sherbiny, and A. A. M. Arafa, “On the solution of the fractional nonlinear
Schrödinger equation,” Physics Letters A, vol. 372, no. 5, pp. 553–558, 2008.
42 I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional
Differential Equations, Some Methods of Their Solution and Some of Their Applications to Methods of Their
Solution and Some of Their Applications, Academic Press, New York, NY, USA, 1999.
43 Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylor’s formula,” Applied Mathematics and
Computation, vol. 186, no. 1, pp. 286–293, 2007.
44 A. A. Kilbas, M. Rivero, L. Rodrı́guez-Germá, and J. J. Trujillo, “α-analytic solutions of some linear
fractional differential equations with variable coefficients,” Applied Mathematics and Computation, vol.
187, no. 1, pp. 239–249, 2007.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014